層論

前言

　　This book is primarily concerned with the study of cohomology theories ofgeneral topological spaces with“general coefficient systems.’’Sheaves playseveral roles in this study.For example.they provide a suitable notion of“general coefl~cient systems.”Moreover.they furnish US with a commonmethod of defining various cohomology theories and of comparison betweendifferent cohomology theories.　　The parts of the theory of sheaves covered here are those areas impor.tant to algebraic topology.Sheaf theory is also important in other fields ofmathematics，notably algebraic geometry，but that iS outside the scope ofthe present book.Thus a more descriptive title for this book might havebeen Algebraic Topology b-om the Point View ol Sheaf Theory.　　Several innovations will be found in this book.　Notably，the con.cept of the“gautness’’of a subspace ran adaptation of an analogous no.tion of Spanier to sheaf-theoretic cohomologyl iS introduced and exploitedthroughout the book.The fact that sheaf-theoretic cohomology satisfiesthe homotopy property is proved for general topological spaces.1 Also，relative cohomology iS introduced into sheaf theory.Concerning relativecohomology，it should be noted that sheaf-theoretic cohomology iS usuallyconsidered as a“single space”theory.This is not without reason.sincecohomology relative to a closed subspace can be obtained by taking coef.ficients in a certain type of sheaf，while that relative to an open subspace（or，more generally，to a taut subspace）can be obtained by taking coho-mology with respect to a special family of supports.However，even in thesecases.it iS sometimes of notational advantage to have a relative cohomologytheory.For example，in our treatment of characteristic classes in ChapterIV the use of relative cohomology enables US to develop the theory in fullgenerality and with relatively simple notation.Our definition of relativecohomology in sheaf theory is the first fully satisfactory one to be given.It is of interest to note that.unlike absolute cohomology，the relative CO-homology groups are not the derived functors of the relative cohomologygroup in degree zero（but they usually are SO in most cases of interest）.

內(nèi)容概要

本書主要講述具有一般系數(shù)體系拓撲空間的上同調(diào)理論。層論包括對代數(shù)拓撲很重要的領(lǐng)域。書中有好多創(chuàng)新點，引進不少新概念，全書內(nèi)容貫穿一致。證實了廣義同調(diào)空間中層理論上同調(diào)滿足同調(diào)基本特性的事實。將相對上同調(diào)引入層理論中?！　∽x者有一定的基本同調(diào)代數(shù)和代數(shù)拓撲知識就可以理解本書。每章末都附有練習，這些可以幫助學生更好的理解書中的知識體系。附錄給出了部分習題的解答。第二版中在內(nèi)容上做了較大的改動，增加了80多例子和大量更深層次的內(nèi)容，如，Cech上同調(diào)、Oliver變換、插值理論、廣義流形、局部齊性空間、同調(diào)纖維和p進變換群。目次：層和準層；層上同調(diào)；與其他上同調(diào)定理的比較；譜序列的應(yīng)用；Borel-Moore同調(diào)；上層和ech同調(diào)?！　∽x者對象：數(shù)學專業(yè)的高年級本科生、研究生和相關(guān)專業(yè)的學者。

書籍目錄

Preface I Sheaves and Presheaves 　1　Definitions 　2　Homomorphisms, subsheaves, and quotient sheaves 　3　Direct and inverse images 　4　Cohomomorphisms 　5　Algebraic constructions 　6　Supports 　7　Classical cohomology theories 　Exercises II Sheaf Cohomology 　I Differential sheaves and resolutions 　2 The canonical resolution and sheaf cohomology 　3 Injective sheaves 　4 Acyclic sheaves 　5 Flabby sheaves 　6 Connected sequences of functors 　7 Axioms for cohomology and the cup product 　8 Maps of spaces 　9 φ-soft and φ-fine sheaves 　10 Subspaces 　11 The Vietoris mapping theorem and homotopy invariance 　12 Relative cohomology 　13 Mayer-Vietoris theorems 　14 Continuity 　15 The Kiinneth and universal coefficient theorems 　16 Dimension 　17 Local connectivity 　18 Change of supports; local cohomology groups 　19 The transfer homomorphism and the Smith sequences 　20 Steenrod's cyclic reduced powers 　21 The Steenrod operations 　Exercises III Comparison with Other Cohomology Theories 　1 Singular cohomology 　2 Alexander-Spanier cohomology 　3 de Rham cohomology 　4 Cech cohomology 　Exercises IV Applications of Spectral Sequerices 　I The spectral sequence of a differential sheaf 　2 The fundamental theorems of sheaves 　3 Direct image relative to a support family 　4 The Leray sheaf 　5 Extension of a support family by a family on the base space 　6 The Leray spectral sequence of a map 　7 Fiber bundles 　8 Dimension 　9 The spectral sequences of Borel and Caftan 　10 Characteristic classes 　11 The spectral sequence of a filtered differential sheaf 　12 The Fary spectral sequence 　13 Sphere bundles with singularities 　14 The Oliver transfer and the Conner conjecture 　Exercises V Borel-Uoore Homology 　I Cosheaves 　2 The dual of a differential cosheaf 　3 Homology theory 　4 Maps of spaces 　5 Subspaces and relative homology 　6 The Vietoris theorem, homotopy, and covering spaces 　7 The homology sheaf of a map 　8 The basic spectral sequences 　9 Poincare duality 　10 The cap product 　11 Intersection theory 　12 Uniqueness theorems 　13 Uniqueness theorems for maps and relative homology 　14 The Kuinneth formula 　15 Change of rings 　16 Generalized manifolds 　17 Locally homogeneous spaces 　18 Homological fibrations and p-adic transformation groups 　19 The transfer homomorphism in homology 　20 Smith theory in homology 　Exercises VI Cosheaves and Cech Homology 　I Theory of cosheaves 　2 Local triviality 　3 Local isomorphisms 　4 Cech homology 　5 The reflector 　6 Spectral sequences 　7 Coresolutions 　8 Relative Cech homology 　9 Locally paracompact spaces 　10 Borel-Moore homology 　11 Modified Borel-Moore homology 　12 Singular homology 　13 Acyclic coverings 　14 Applications to maps 　Exercises A Spectral Sequences 　1 The spectral sequence of a filtered complex 　2 Double complexes 　3 Products 　4 Homomorphisms B Solutions to Selected Exercises 　Solutions for Chapter I 　Solutions for Chapter II 　Solutions for Chapter III 　Solutions for Chapter IV 　Solutions for Chapter V 　Solutions for Chapter VI Bibliography List of Symbols List of Selected Facts Index

章節(jié)摘錄

　　In this chapter we shall define the sheaf-theoretic cohomology theory andshall develop many of its basic properties.　　The cohomology groups of a space with coefficients in a sheaf are definedin Section 2 using the canonical resolution of a sheaf due to Godement.In Section 3 it is shown that the category of sheaves contains "enoughinjectives," and it follows from the results of Sections 4 and 5 that thesheaf cohomology groups are just the right derived functors of the leftexact functor F that assigns to a sheaf its group of sections.　　A sheaf is is said to be acyclic if the higher cohomology groups withcoefficients in d are zero. Such sheaves provide a means of "computing"cohomology in particular situations. In Sections 5 and 9 some importantclasses of acyclic sheaves are defined and investigated.　　In Section 6 we prove a theorem concerning the existence and uniquenessof extensions of a natural transformation of functors （of several variables）to natural transformations of "connected systems" of functors. This resultis applied in Section 7 to define, and to give axioms for, the cup productin sheaf cohomology theory. These sections are central to our treatment ofmany of the fundamental consequences of sheaf theory.　　The cohomology homomorphism induced by a map is defined in Section8. The relationship between the cohomology of a subspace and that of itsneighborhoods is investigated in Section 10, and the important notion of"tautness" of a subspace is introduced there.　　In Section 11 we prove the Vietoris mapping theorem and use it toprove that sheaf-theoretic cohomology, with constant coefficients, satisfiesthe invariance under homotopy property for general topological spaces.　　Relative cohomology theory is introduced into sheaf theory in Section12, and its properties, such as invariance under excision, are developed. InSection 13 we derive some exact sequences of the Mayer-Vietoris type.

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